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Suppose we are given a fiber bundle $p:E \to B$ and a point $x \in B$. Denote by $p \big|_{p^{-1}(x)}:p^{-1}(x) \to B$ the restricted fiber bundle and by $\Gamma^0(p)$ (resp. $\Gamma^0(p \big|_{p^{-1}(x)})$) the space of continuous sections with the compact-open topology. Is the restriction map $\rho:\Gamma^0(p)\to\Gamma^0(p\big|_{p^{-1}(x)})$ in general a weak homotopy equivalence or not? I'm obviously having trouble with some basic things but any help is well appreciated.

UPDATE: Just to make clear what I have so far: since we're considering the restriction to one point we have that $\Gamma^0(p\big|_{p^{-1}(x)})\cong p^{-1}(x)$ and thus it seems to me highly unlikely that $\rho$ might be a weak homotopy equivalence in general. Except if I'm missing some very basic fact about fiber bundles which is what I'm here to check for.

UPDATE 2: I just tried to prove surjectivity of the induced map $\rho_*$ on homotopy in a general setting but was unsuccessful, hinting at either a wrong assumption or an inept mathematician.

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@willie Thanks for the suggestion, it looks like the software (or more likely some astute moderator) realized that the unregistered Martin and the newly registered Martin are the same. –  Martin Worsek Jun 28 '11 at 15:01
    
that was me. A good samaritan was nice enough to ping me when he noticed that you have both a registered and an unregistered account. –  Willie Wong Jun 29 '11 at 9:50
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The statement in the original paragraph is very obviously not true in general. Instead assuming additionally contractibility of the base space one can very easily show that restriction-to-a-fiber/evaluation maps of sections do constitute weak homotopy equivalences. Thanks go to Dan Ramras for pointing that out to me.

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